Newform orbit 77.2.e.b

• Label: 77.2.e.b
• Level: $77$
• Weight: $2$
• Character orbit: 77.e
• Analytic conductor: $0.615$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.614848095564\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.1783323.2
Defining polynomial:	\( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b5 - b3) * q^2 + b1 * q^3 + (b5 + b4 - b1 - 1) * q^4 + (-b5 + b4 + b3 + b2 + b1) * q^5 + (-b3 + b2) * q^6 + (-b5 - b4 + b3 - b1 + 1) * q^7 + (b3 - 3) * q^8 + (b5 - b3) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} - 4 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} - 18 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 19\nu^{2} + 12\nu - 60 ) / 83 \)
\(\beta_{3}\)	\(=\)	\( ( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 48\nu - 240 ) / 83 \)
\(\beta_{4}\)	\(=\)	\( ( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu + 204 ) / 249 \)
\(\beta_{5}\)	\(=\)	\( ( -16\nu^{5} - 3\nu^{4} - 68\nu^{3} - 28\nu^{2} - 275\nu - 36 ) / 83 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(-\beta_{4}\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

23.1

1.09935	+	1.90412i
−0.956115	−	1.65604i
0.356769	+	0.617942i
1.09935	−	1.90412i
−0.956115	+	1.65604i
0.356769	−	0.617942i

−0.917122

+

1.58850i

1.09935

+

1.90412i

−0.682224

−

1.18165i

0.317776

−

0.550404i

−4.03293

0.317776

−

2.62660i

−1.16576

−0.917122

+

1.58850i

0.582878

+

1.00958i

23.2

−0.328310

+

0.568650i

−0.956115

−

1.65604i

0.784425

+

1.35866i

1.78442

−

3.09071i

1.25561

1.78442

+

1.95341i

−2.34338

−0.328310

+

0.568650i

1.17169

+

2.02943i

23.3

1.24543

−

2.15715i

0.356769

+

0.617942i

−2.10220

−

3.64112i

−1.10220

+

1.90907i

1.77733

−1.10220

+

2.40523i

−5.49086

1.24543

−

2.15715i

2.74543

+

4.75523i

67.1

−0.917122

−

1.58850i

1.09935

−

1.90412i

−0.682224

+

1.18165i

0.317776

+

0.550404i

−4.03293

0.317776

+

2.62660i

−1.16576

−0.917122

−

1.58850i

0.582878

−

1.00958i

67.2

−0.328310

−

0.568650i

−0.956115

+

1.65604i

0.784425

−

1.35866i

1.78442

+

3.09071i

1.25561

1.78442

−

1.95341i

−2.34338

−0.328310

−

0.568650i

1.17169

−

2.02943i

67.3

1.24543

+

2.15715i

0.356769

−

0.617942i

−2.10220

+

3.64112i

−1.10220

−

1.90907i

1.77733

−1.10220

−

2.40523i

−5.49086

1.24543

+

2.15715i

2.74543

−

4.75523i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	77.2.e.b	✓	6
3.b	odd	2	1		693.2.i.g		6
4.b	odd	2	1		1232.2.q.k		6
7.b	odd	2	1		539.2.e.l		6
7.c	even	3	1	inner	77.2.e.b	✓	6
7.c	even	3	1		539.2.a.h		3
7.d	odd	6	1		539.2.a.i		3
7.d	odd	6	1		539.2.e.l		6
11.b	odd	2	1		847.2.e.d		6
11.c	even	5	4		847.2.n.e		24
11.d	odd	10	4		847.2.n.d		24
21.g	even	6	1		4851.2.a.bn		3
21.h	odd	6	1		693.2.i.g		6
21.h	odd	6	1		4851.2.a.bo		3
28.f	even	6	1		8624.2.a.ck		3
28.g	odd	6	1		1232.2.q.k		6
28.g	odd	6	1		8624.2.a.cl		3
77.h	odd	6	1		847.2.e.d		6
77.h	odd	6	1		5929.2.a.v		3
77.i	even	6	1		5929.2.a.w		3
77.m	even	15	4		847.2.n.e		24
77.o	odd	30	4		847.2.n.d		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.e.b	✓	6	1.a	even	1	1	trivial
77.2.e.b	✓	6	7.c	even	3	1	inner
539.2.a.h		3	7.c	even	3	1
539.2.a.i		3	7.d	odd	6	1
539.2.e.l		6	7.b	odd	2	1
539.2.e.l		6	7.d	odd	6	1
693.2.i.g		6	3.b	odd	2	1
693.2.i.g		6	21.h	odd	6	1
847.2.e.d		6	11.b	odd	2	1
847.2.e.d		6	77.h	odd	6	1
847.2.n.d		24	11.d	odd	10	4
847.2.n.d		24	77.o	odd	30	4
847.2.n.e		24	11.c	even	5	4
847.2.n.e		24	77.m	even	15	4
1232.2.q.k		6	4.b	odd	2	1
1232.2.q.k		6	28.g	odd	6	1
4851.2.a.bn		3	21.g	even	6	1
4851.2.a.bo		3	21.h	odd	6	1
5929.2.a.v		3	77.h	odd	6	1
5929.2.a.w		3	77.i	even	6	1
8624.2.a.ck		3	28.f	even	6	1
8624.2.a.cl		3	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 5T_{2}^{4} + 6T_{2}^{3} + 25T_{2}^{2} + 15T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^6 + 5*T^4 + 6*T^3 + 25*T^2 + 15*T + 9
$3$	T^6 - T^5 + 5*T^4 - 2*T^3 + 19*T^2 - 12*T + 9
$5$	T^6 - 2*T^5 + 11*T^4 + 4*T^3 + 59*T^2 - 35*T + 25
$7$	T^6 - 2*T^5 + 14*T^4 - 23*T^3 + 98*T^2 - 98*T + 343
$11$	\( (T^{2} - T + 1)^{3} \)